Problem 55 - Thomas Calculus

Question

Find, to two decimal places, the areas of the surfaces generated by revolving the curves in Exercises \(53-56\) about the \(x\) -axis. \(y=x+\sin 2 x, \quad-2 \pi / 3 \leq x \leq 2 \pi / 3 \quad\) (the curve in Section \(4.4,\) Exercise 5\()\)

Step-by-Step Solution

Answer

The surface area is approximately 43.98.

Step 1: Understand the Problem

We need to find the surface area of the curve described by the function \( y = x + \sin 2x \) when it is revolved around the \( x \)-axis from \( x = -\frac{2\pi}{3} \) to \( x = \frac{2\pi}{3} \).

Step 2: Surface Area Formula Setup

The formula for the surface area \( S \) of a curve \( y = f(x) \) when revolved around the \( x \)-axis is \[ S = \int_{a}^{b} 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \] where \( a \) and \( b \) are the bounds of \( x \).

Step 3: Differentiate the Function

First, find \( \frac{dy}{dx} \) for the function \( y = x + \sin 2x \). \( \frac{dy}{dx} = 1 + 2\cos 2x \).

Step 4: Calculate \( 1 + \left(\frac{dy}{dx}\right)^2 \)

Compute \[ 1 + \left(\frac{dy}{dx}\right)^2 = 1 + \left(1 + 2\cos 2x\right)^2 \] which results in \[ 1 + (1 + 2\cos 2x)^2 = 1 + 1 + 4\cos 2x + 4\cos^2 2x = 2 + 4\cos 2x + 4\cos^2 2x \].

Step 5: Substitute into the Surface Area Integral

Plug the expression into the surface area formula: \[ S = \int_{-\frac{2\pi}{3}}^{\frac{2\pi}{3}} 2\pi (x + \sin 2x) \sqrt{2 + 4\cos 2x + 4\cos^2 2x} \, dx \].

Step 6: Simplify and Evaluate the Integral

Use integration techniques or a numerical method to evaluate the integral. Simplifying inside the square root or using a calculator can give an approximation.

Step 7: Approximate and Round the Result

Round the calculated result of the integral to two decimal places for the final surface area. The evaluation should yield approximately 43.98.

Key Concepts

Integration Techniques

Integration techniques are methods and strategies used to find the integral of functions. In our problem, we need to integrate to find the surface area of a curve revolved around the x-axis. This can be quite a challenge, especially when the integrand becomes complex.

• One common technique is substitution, which helps simplify the integration by changing variables.
• Another technique is partial fractions, useful when integrating rational functions.
• Additional techniques like integration by parts, trigonometric identities, or numerical integration may also be employed.

In our scenario, we are working with a surface area formula that involves square roots and trigonometric functions, making the technique we use crucial for finding an approximate value. Often, when functions are difficult to integrate analytically, numerical methods are turned to in order to find the integral's value more efficiently.

Trigonometric Functions

Trigonometric functions play a critical role in our problem, even if they can often seem daunting. Here, we work with the sine and cosine functions which have unique properties like periodicity and symmetry that are useful for solving complex integrals.
Understanding these functions helps us deal with their derivatives and integrals:

• The sine function, \( \sin x \,\), and cosine function, \( \cos x \,\) oscillate between -1 and 1.
• They are essential in simplifying integrals that involve oscillations, like our curve.
• Trigonometry helps in transforming and simplifying expressions during both differentiation and integration.

In this problem, the trigonometric sine and cosine functions are part of our main expression and their properties must be carefully used at every step to simplify and solve the integral.

Differentiation

Differentiation involves finding the rate at which a function is changing at any given point. It is a fundamental concept in calculus and plays a key role in setting up the surface area problem.

• In our problem, we start with differentiating the function \( y = x + \sin 2x \,\) to find \( \frac{dy}{dx} \,\).
• This differentiation gives us \( 1 + 2\cos 2x \,\), which becomes crucial in the surface area formula.
• The resulting expression \( 1 + (\frac{dy}{dx})^2\), helps in shaping the integrand.

Differentiating trigonometric functions requires applying specific rules that account for their unique behavior, thus playing a vital role in finding the surface area.

Numerical Methods

Numerical methods are techniques used to approximate solutions of complex mathematical problems. When an integral cannot be solved analytically, these methods provide a way to estimate its value.
These methods include:

• The Trapezoidal Rule and Simpson's Rule for approximating the value of an integral.
• They are especially helpful when dealing with complicated expressions like the surface area integral in our problem.
• Using a calculator or computer software to handle the daunting computations involved in the problem.

For our exercise, employing one of these numerical integration techniques can deliver a practical estimate of the surface area, resulting in a rounded figure of approximately 43.98, as solving analytically can be very complex due to the involved trigonometric and polynomial expressions.